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Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$ whose adelicGalois representation has open image in $\text{GSp}_{2g} \widehat{\mathbb{Z}}$.We investigate the endomorphism algebras $\text{End}(A_p) \otimes \mathbb{Q} =\mathbb{Q}( \pi_p )$ of the reduction of $A$ modulo primes $p$ at which thisreduction is ordinary and simple. We obtain conditional and unconditionalasymptotic upper bounds on the number of primes at which this "Frobenius field"is a specified number field and, when $A$ is two-dimensional, at which theFrobenius field contains a specified real quadratic number field. Theseinvestigations continue the investigations of variants of the Lang-TrotterConjectures on elliptic curves.

The Square Sieve and a Lang-Trotter Question for Generic Abelian Varieties